Products of Random Matrices and Generalised Quantum Point Scatterers

To every product of 2×2 matrices, there corresponds a one-dimensional Schrödinger equation whose potential consists of generalised point scatterers. Products of random matrices are obtained by making these interactions and their positions random. We exhibit a simple one-dimensional quantum model corresponding to the most general product of matrices in SL(2,ℝ). We use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms.